Two efficient values of cooperative games with graph structure based on $$\tau $$-values

The paper is devoted to value concepts for cooperative games with a communication structure represented by a graph. Under assumptions that the players partition themselves into ‘components’ before realizing cooperation and the worth of the grand coalition not less than the sum of the worths of all components, the fair distribution of surplus solution and the two-step \(\tau \)-value are introduced as two efficient values for such games, each of which is an extension of the graph \(\tau \)-value. For the two efficient values, we discuss their special properties and we provide their axiomatic characterizations in views of those properties. By analysing an example applied to the two values, we conclude that the fair distribution of surplus solution allocates more surplus to the bigger coalitions and favors the powerful players, while the two-step \(\tau \)-value benefits the vulnerable groups and inspires to form small coalitions.     

The Farsighted Stable Set

Harsanyi (1974) criticized the von Neumann–Morgenstern (vNM) stable set for its presumption that coalitions are myopic about their prospects. He proposed a new dominance relation incorporating farsightedness, but retained another feature of the stable set: that a coalition S can impose any imputation as long as its restriction to S is feasible for it. This implicitly gives an objecting coalition complete power to arrange the payoffs of players elsewhere, which is clearly unsatisfactory. While this assumption is largely innocuous for myopic dominance, it is of crucial significance for its farsighted counterpart. Our modification of the Harsanyi set respects “coalitional sovereignty.” The resulting farsighted stable set is very different from both the Harsanyi and the vNM sets. We provide a necessary and sufficient condition for the existence of a farsighted stable set containing just a single‐payoff allocation. This condition roughly establishes an equivalence between core allocations and the union of allocations over all single‐payoff farsighted stable sets. We then conduct a comprehensive analysis of the existence and structure of farsighted stable sets in simple games. This last exercise throws light on both single‐payoff and multi‐payoff stable sets, and suggests that they do not coexist.     

基于合作博弈模型的公共河流水资源分配方案研究

水资源的稀缺性使得公共河流的水资源分配问题中存在着各种冲突与矛盾。为了确定公共河流流经的各用水主体之间分配水资源的合理方案，本文建立具有外部性的合作博弈模型来分析公共河流流经的各用水主体之间在竞争与合作并存情况下的水资源分配问题。使用动态博弈的方法来确定联盟之间通过竞争产生的均衡水资源分配量，然后在各联盟之内通过Nash协商的方式来分配联盟的均衡水资源分配量，比较各种方案下用水主体产生的总效用，进而得到公共河流用水主体之间竞争与合作并存时的最优水资源分配方案和各用水主体之间形成联盟的具体形式。研究表明：受到外部性环境的影响，公共河流用水主体之间部分合作可能比完全合作产生更大的总效用，合作与竞争并存时的最优分配方案优于完全合作时的最优分配方案。     

具有联盟优先关系的模糊合作博弈的目标规划求解模型

用合作博弈研究实际管理问题中的分配方案时，常常存在一些不重要联盟或无效联盟，这些联盟影响公平合理的分配方案。因此，联盟的重要程度成为求解合作博弈必不可少的因素。本文考虑了联盟的重要性和局中人参与联盟的不确定性，研究了具有优先关系的模糊联盟合作博弈(简称为模糊合作博弈)。首先，借助于目标规划模型的优先因子可以表征联盟重要程度的思想，通过构建多优先级目标规划模型，得到模糊合作博弈新的解。其次，证明了构建的多优先级目标规划模型的解和模糊合作博弈的核心之间具有重要对应关系。最后，通过数值实例和比较分析，说明本文提出的多优先级目标规划模型求解模糊合作博弈问题的合理性和有效性。研究表明：(1)本文提出的多优先级目标规划模型考虑不同联盟重要程度，得到的解符合“多劳多得”原则，能够更公平合理解决实际管理中的分配问题。(2)本文的目标规划模型同时适用于求解存在联盟特征函数值缺失的合作博弈。与已有合作博弈的解进行比较分析，该模型无需估算无效联盟的特征函数缺失数据得到的分配值更为准确。从而，说明本文给出的目标规划求解模糊合作博弈解的模型，更符合许多管理学问题的实际情况。(3)通过多优先级目标规划模型最优解是否存在可判断模糊合作博弈的核心存在情况，若核心存在则该模型通过目标规划软件包可得到核心内的一个解，这样也得到了一个判断合作博弈核心是否存在的标准。(4)目标规划模型可弥补已有合作博弈解的一些不足，如核心可能为空集，Shapley值和最小二乘预核仁可能不满足个体合理性等。本文构建的多优先级目标规划模型不仅能求解联盟具有优先关系的模糊合作博弈，而且能够求解一般合作博弈的解，该目标规划模型作为合作博弈一种新的求解方法，能更有效地解决实际管理中的分配问题，具有更加广泛的应用价值。     

Solving constrained optimization problems by solution-based decomposition search

Solving constrained optimization problems (COPs) is a challenging task. In this paper we present a new strategy for solving COPs called solve and decompose (or \( S \& D\) for short). The proposed strategy is a systematic iterative depth-first strategy that is based on problem decomposition. \( S \& D\) uses a feasible solution of the COP, found by any exact method, to further decompose the original problem into a bounded number of subproblems which are considerably smaller in size. It also uses the value of the feasible solution as a bound that we add to the created subproblems in order to strengthen the cost-based filtering. Furthermore, the feasible solution is exploited in order to create subproblems that have more promise in finding better solutions which are explored in a depth-first manner. The whole process is repeated until we reach a specified depth where we do not decompose the subproblems anymore but we solve them to optimality using any exact method like Branch and Bound. Our initial results on two benchmark problems show that \( S \& D\) may reach improvements of up to three orders of magnitude in terms of runtime when compared to Branch and Bound.     

Political Economy of Redistribution

It is often argued that additional constraints on redistribution such as granting veto power to more players in society better protects property from expropriation. We use a model of multilateral bargaining to demonstrate that this intuition may be flawed. Increasing the number of veto players or raising the supermajority requirement for redistribution may reduce protection on the equilibrium path. The reason is the existence of two distinct mechanisms of property protection. One is formal constraints that allow individuals or groups to block any redistribution that is not in their favor. The other occurs in equilibrium where players without such powers protect each other from redistribution. Players without formal veto power anticipate that the expropriation of other similar players will ultimately hurt them and thus combine their influence to prevent redistributions. In a stable allocation, the society exhibits a “class” structure with class members having equal wealth and strategically protecting each other from redistribution.     

Benefactors and Beneficiaries: The Effects of Giving and Receiving on Cost‐Coalitional Problems

Motivated by supply chain collaborations in practice, we introduce a class of cost‐coalitional problems, which are based on a priori information about the cost faced by each agent in each set that it could belong to. Our focus is on problems with decreasingly monotonic coalitional costs. In this class of problems, we study the effects of giving and receiving when there exist players whose participation in an alliance always contributes to the savings of all alliance members (we refer to these players as benefactors), and there also exist players whose cost decreases in such an alliance (we call them beneficiaries). We use linear and quadratic norm cost games to analyze the role played by benefactors and beneficiaries in achieving stability of different cooperating alliances. We consider different notions of stability (the core and the bargaining set) and provide conditions for stability of an all‐inclusive alliance of agents which leads to minimum value of total cost incurred by all agents.     

Axiomatic Equilibrium Selection for Generic Two‐Player Games

For a finite game with perfect recall, a refinement of its set of Nash equilibria selects closed connected subsets, called solutions. Assume that each solution's equilibria use undominated strategies and some of its equilibria are quasi‐perfect, and that all solutions are immune to presentation effects; namely, if the game is embedded in a larger game with more pure strategies and more players such that the original players' feasible mixed strategies and expected payoffs are preserved regardless of what other players do, then the larger game's solutions project to the original game's solutions. Then, for a game with two players and generic payoffs, each solution is an essential component of the set of equilibria that use undominated strategies, and thus a stable set of equilibria as defined by Mertens (1989).     

The QAP-polytope and the graph isomorphism problem

In this paper we propose a geometric approach to solve the Graph Isomorphism (GI in short) problem. Given two graphs \(G_1, G_2\), the GI problem is to decide if the given graphs are isomorphic i.e., there exists an edge preserving bijection between the vertices of the two graphs. We propose an Integer Linear Program (ILP) that has a non-empty solution if and only if the given graphs are isomorphic. The convex hull of all possible solutions of the ILP has been studied in literature as the Quadratic Assignment Problem (QAP) polytope. We study the feasible region of the linear programming relaxation of the ILP and show that the given graphs are isomorphic if and only if this region intersects with the QAP-polytope. As a consequence, if the graphs are not isomorphic, the feasible region must lie entirely outside the QAP-polytope. We study the facial structure of the QAP-polytope with the intention of using the facet defining inequalities to eliminate the feasible region outside the polytope. We determine two new families of facet defining inequalities of the QAP-polytope and show that all the known facet defining inequalities are special instances of a general inequality. Further we define a partial ordering on each exponential sized family of facet defining inequalities and show that if there exists a common minimal violated inequality for all points in the feasible region outside the QAP-polytope, then we can solve the GI problem in polynomial time. We also study the general case when there are k such inequalities and give an algorithm for the GI problem that runs in time exponential in k.     

价值创造结构、K-重划分与联盟稳定性变化研究

与传统以“价值增加”为核心的边际分析不同,强调价值创造“结构”本身的特性.认为联盟结构的特征函数集合描述了价值创造的力量结构和联盟是否稳定的内在属性.研究表明,价值创造的特征函数结构同联盟核存在的最小价值创造的值之间存在线性变换关系,联盟的最大的k-重划分值等于联盟核存在的最小值.在联盟稳定性变化的临界状态下,其划分点作为了联盟结构的“合并”和“分解”点,不能出现在交集为空的互补性集合之间,各子联盟集合满足价值创造非互补性和线性可加性.     

Reconfiguration of dominating sets

We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph G is a set S of vertices such that each vertex is either in S or has a neighbour in S. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions s and t such that each pair of consecutive solutions is adjacent according to a specified adjacency relation. Two dominating sets are adjacent if one can be formed from the other by the addition or deletion of a single vertex. For various values of k, we consider properties of \(D_k(G)\), the graph consisting of a node for each dominating set of size at most k and edges specified by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that \(D_{\varGamma (G)+1}(G)\) is not necessarily connected, for \(\varGamma (G)\) the maximum cardinality of a minimal dominating set in G. The result holds even when graphs are constrained to be planar, of bounded tree-width, or b-partite for \(b \ge 3\). Moreover, we construct an infinite family of graphs such that \(D_{\gamma (G)+1}(G)\) has exponential diameter, for \(\gamma (G)\) the minimum size of a dominating set. On the positive side, we show that \(D_{n-\mu }(G)\) is connected and of linear diameter for any graph G on n vertices with a matching of size at least \(\mu +1\).     

Improved algorithmic results for unsplittable stable allocation problems

The stable allocation problem is a many-to-many generalization of the well-known stable marriage problem, where we seek a bipartite assignment between, say, jobs (of varying sizes) and machines (of varying capacities) that is “stable” based on a set of underlying preference lists submitted by the jobs and machines. Building on the initial work of Dean et al. (The unsplittable stable marriage problem, 2006), we study a natural “unsplittable” variant of this problem, where each assigned job must be fully assigned to a single machine. Such unsplittable bipartite assignment problems generally tend to be NP-hard, including previously-proposed variants of the unsplittable stable allocation problem (McDermid and Manlove in J Comb Optim 19(3): 279–303, 2010). Our main result is to show that under an alternative model of stability, the unsplittable stable allocation problem becomes solvable in polynomial time; although this model is less likely to admit feasible solutions than the model proposed in McDermid and Manlove (J Comb Optim 19(3): 279–303, McDermid and Manlove 2010), we show that in the event there is no feasible solution, our approach computes a solution of minimal total congestion (overfilling of all machines collectively beyond their capacities). We also describe a technique for rounding the solution of a stable allocation problem to produce “relaxed” unsplit solutions that are only mildly infeasible, where each machine is overcongested by at most a single job.     

On revenue maximization with sharp multi-unit demands

We consider markets consisting of a set of indivisible items, and buyers that have sharp multi-unit demand. This means that each buyer \(i\) wants a specific number \(d_i\) of items; a bundle of size less than \(d_i\) has no value. We consider the objective of setting prices and allocations in order to maximize the total revenue of the market maker. The pricing problem with sharp multi-unit demand buyers has a number of properties that the unit-demand model does not possess, and is an important question in algorithmic pricing. We consider the problem of computing a revenue maximizing solution for two solution concepts: competitive equilibrium and envy-free pricing. For unrestricted valuations, these problems are NP-complete; we focus on a realistic special case of “correlated values” where each buyer \(i\) has a valuation \(v_iq_j\) for item \(j\), where \(v_i\) and \(q_j\) are positive quantities associated with buyer \(i\) and item \(j\) respectively. We present a polynomial time algorithm to solve the revenue-maximizing competitive equilibrium problem. For envy-free pricing, if the demand of each buyer is bounded by a constant, a revenue maximizing solution can be found efficiently; the general demand case is shown to be NP-hard.     

Degree-constrained orientations of embedded graphs

We investigate the problem of orienting the edges of an embedded graph in such a way that the resulting digraph fulfills given in-degree specifications both for the vertices and for the faces of the embedding. This primal-dual orientation problem was first proposed by Frank for the case of planar graphs, in conjunction with the question for a good characterization of the existence of such orientations. We answer this question by showing that a feasible orientation of a planar embedding, if it exists, can be constructed by combining certain parts of a primally feasible orientation and a dually feasible orientation. For the general case of arbitrary embeddings, we show that the number of feasible orientations is bounded by \(2^{2g}\), where \(g\) is the genus of the embedding. Our proof also yields a fixed-parameter algorithm for determining all feasible orientations parameterized by the genus. In contrast to these positive results, however, we also show that the problem becomes \(N\!P\)-complete even for a fixed genus if only upper and lower bounds on the in-degrees are specified instead of exact values.     

非合作-合作两型博弈的Shapley值纯策略纳什均衡解求解方法

在合作中又有竞争的"经济全球化"时代背景下,经济实体之间越来越多地体现出竞争与合作交织的特点,既有策略的选择,同时也有利益的分配或者成本的分摊,即竞争与合作相互联系。为此,Brandenburger和Stuart提出了非合作-合作两型博弈模型为这类博弈提供了有效的工具。目前非合作-合作两型博弈研究较少,且Brandenburger和Stuart提出的非合作-合作两型博弈存在一些不足：合作博弈用核心求解可能为空或者不唯一。Shapley值是一种重要的合作博弈单值解,满足匿名性、有效性、可加性和虚拟性,表达形式简单且唯一,对一些成本分摊问题和利益分配问题,给决策者提供了一个公平满意的分配方案。因此本文研究将Shapley值作为合作博弈的解时非合作-合作两型博弈解存在的条件。为了分析本文提出的基于Shapley值的非合作-合作两型博弈的新理论框架,首先给出了其特征函数满足的联盟无外部性条件。在满足此条件下,我们进一步证明了非合作-合作两型博弈解存在的条件及性质。结合数值实例比较分析合作博弈用核心和Shapley值求解非合作-合作两型博弈解的优缺点。研究表明：当用Shapley值求解合作博弈解,降低了非合作-合作两型博弈解存在条件。因此,本文的研究不仅弥补了Brandenburger和Stuart提出的非合作-合作两型博弈中合作博弈的核心为空或者不唯一的情况,而且为非合作-合作两型博弈的解提供新的理论框架,从而为既有竞争又有合作的博弈问题提供新的求解方法,因此,本文的研究具有一定的理论价值和应用价值。     

Online set multicover algorithms for dynamic D2D communications

Motivated by the dynamic resource allocation problem for device-to-device (D2D) communications, we study the online set multicover problem (OSMC). In the online set multicover, the set X of elements to be covered is unknown in advance; furthermore, the coverage requirement of each element \(x \in X\) is initially unknown. Elements of X together with coverage requirements are presented one at a time in an online fashion; and a feasible solution must be maintained at all times. We provide the first deterministic, online algorithms for OSMC with competitive ratios. We consider two versions of OSMC; in the first, each set may be picked only once, while the second version allows each set to be picked multiple times. For both versions, we present the first deterministic, online algorithms, with competitive ratios \(O( \log n \log m )\) and \(O( \log n (\log m + \log k) )\), repectively, where n is the number of elements, m is the number of sets, and k is the maximum coverage requirement. By simulation, we show the efficacy of these algorithms for resource allocation in the D2D setting by analyzing network throughput and other metrics, obtaining a large improvement in running time over offline methods.     

Scheduling dedicated jobs with variative processing times

We consider the following optimization problem. There is a set of \(n\) dedicated jobs that are to be processed on \(m\) parallel machines. The job set is partitioned into subsets and jobs of each subset have a common due date. Processing times of jobs are interconnected and they are the subject of the decision making. The goal is to choose a processing time for each job in a feasible way and to construct a schedule that minimizes the maximum lateness. We show that the problem is NP-hard even if \(m=1\) and that it is NP-hard in the strong sense if \(m\) is a variable. We prove that there is no approximate polynomial algorithm with guaranteed approximation ratio less than 2. We propose an integer linear formulation for the problem and perform experiments. The experiments show that the solutions obtained with CPLEX within the limit of 5 min are on average about 5 % from the optimum value for instances with up to 150 jobs, 16 subsets and 11 machines. Most instances were solved to optimality and the average CPLEX running time was 32 s for these instances.     

Cost sharing and strategyproof mechanisms for set cover games

We develop for set cover games several general cost-sharing methods that are approximately budget-balanced, in the core, and/or group-strategyproof. We first study the cost sharing for a single set cover game, which does not have a budget-balanced mechanism in the core. We show that there is no cost allocation method that can always recover more than $\frac{1}{\ln n}$ of the total cost and in the core. Here n is the number of all players to be served. We give a cost allocation method that always recovers $\frac{1}{\ln d_{\mathit{max}}}$ of the total cost, where d _max is the maximum size of all sets. We then study the cost allocation scheme for all induced subgames. It is known that no cost sharing scheme can always recover more than $\frac{1}{n}$ of the total cost for every subset of players. We give an efficient cost sharing scheme that always recovers at least $\frac{1}{2n}$ of the total cost for every subset of players and furthermore, our scheme is cross-monotone. When the elements to be covered are selfish agents with privately known valuations, we present a strategyproof charging mechanism, under the assumption that all sets are simple sets; further, the total cost of the set cover is no more than ln?d _max times that of an optimal solution. When the sets are selfish agents with privately known costs, we present a strategyproof payment mechanism to them. We also show how to fairly share the payments to all sets among the elements.     

Managing Imbalanced Supply Chain Relationships for Sustainability: A Power Perspective

This study adopts a power perspective to investigate sustainable supply chain relationships and specifically uses resource‐dependence theory (RDT) to critically analyze buyer–supplier–supplier relationships. Empirical evidence is provided, extending the RDT model in this context. The concept of power relationships is explored through a qualitative study of a multinational company and agricultural growers in the UK food industry that work together to implement sustainable practices. We look at multiple triadic relationships involving a large buyer and its small suppliers to investigate how relative power affects the implementation of sustainable supply‐management practices. The study highlights that power as dependence is relevant to understanding compliance in sustainable supply chains and to identifying appropriate relationship‐management strategies to build more sustainable supply chains. We show the influences of power on how players manage their relationships and how it affects organizational responses to the implementation of sustainability initiatives. Power notably influences the sharing of sustainability‐related risks and value between supply chain partners. From a managerial perspective, the study contributes to developing a better understanding of how power can become an effective way to achieve sustainability goals. This article offers insights into the way in which a large organization works with small and medium size enterprises to implement sustainable practices and shows how power management—that is, the way in which power is used—can support or hinder effective cooperation around sustainability in the supply chain.     

Characterisation of forests with trivial game domination numbers

In the domination game, two players, the Dominator and Staller, take turns adding vertices of a fixed graph to a set, at each turn increasing the number of vertices dominated by the set, until the final set \(A_*\) dominates the whole graph. The Dominator plays to minimise the size of the set \(A_*\) while the Staller plays to maximise it. A graph is \(D\)-trivial if when the Dominator plays first and both players play optimally, the set \(A_*\) is a minimum dominating set of the graph. A graph is \(S\)-trivial if the same is true when the Staller plays first. We consider the problem of characterising \(D\)-trivial and \(S\)-trivial graphs. We give complete characterisations of \(D\)-trivial forests and of \(S\)-trivial forests. We also show that \(2\)-connected \(D\)-trivial graphs cannot have large girth, and conjecture that the same holds without the connectivity condition.